3.1668 \(\int \frac{1}{(a+\frac{b}{x}) \sqrt{x}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]

[Out]

(2*Sqrt[x])/a - (2*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(3/2)

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Rubi [A]  time = 0.013709, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*Sqrt[x]),x]

[Out]

(2*Sqrt[x])/a - (2*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(3/2)

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right ) \sqrt{x}} \, dx &=\int \frac{\sqrt{x}}{b+a x} \, dx\\ &=\frac{2 \sqrt{x}}{a}-\frac{b \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{a}\\ &=\frac{2 \sqrt{x}}{a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0093032, size = 40, normalized size = 1. \[ \frac{2 \sqrt{x}}{a}-\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*Sqrt[x]),x]

[Out]

(2*Sqrt[x])/a - (2*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(3/2)

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{a}}-2\,{\frac{b}{a\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)/x^(1/2),x)

[Out]

2*x^(1/2)/a-2*b/a/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.80438, size = 189, normalized size = 4.72 \begin{align*} \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{a x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \, \sqrt{x}}{a}, -\frac{2 \,{\left (\sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) - \sqrt{x}\right )}}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^(1/2),x, algorithm="fricas")

[Out]

[(sqrt(-b/a)*log((a*x - 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*x + b)) + 2*sqrt(x))/a, -2*(sqrt(b/a)*arctan(a*sqrt(x)*
sqrt(b/a)/b) - sqrt(x))/a]

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Sympy [A]  time = 1.37189, size = 92, normalized size = 2.3 \begin{align*} \begin{cases} \frac{2 \sqrt{x}}{a} + \frac{i \sqrt{b} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} - \frac{i \sqrt{b} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{2} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x**(1/2),x)

[Out]

Piecewise((2*sqrt(x)/a + I*sqrt(b)*log(-I*sqrt(b)*sqrt(1/a) + sqrt(x))/(a**2*sqrt(1/a)) - I*sqrt(b)*log(I*sqrt
(b)*sqrt(1/a) + sqrt(x))/(a**2*sqrt(1/a)), Ne(a, 0)), (2*x**(3/2)/(3*b), True))

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Giac [A]  time = 1.08196, size = 42, normalized size = 1.05 \begin{align*} -\frac{2 \, b \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{2 \, \sqrt{x}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^(1/2),x, algorithm="giac")

[Out]

-2*b*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a) + 2*sqrt(x)/a